Maybe manipulatives aren't the answer?

This NYT article suggests that

... it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations ... Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler ... performed a randomized, controlled experiment. ... Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school ...

In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. ... The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”